Logic, Reasoning / Logic

This category emphasizes general logical reasoning skills, often applied to puzzles or scenarios not strictly formal. It involves deduction, inference, identifying patterns, and drawing sound conclusions from given information. It overlaps with formal logic but can be broader.

• Apples and Pears

  There is a basket containing `30` fruits. It is known that among any `12` fruits we take from the basket, there is necessarily at least one apple, and among any `20` fruits there is necessarily one pear. How many apples and how many pears are there in the basket?

  Topics:

  Combinatorics -> Pigeonhole Principle Algebra -> Inequalities Logic -> Reasoning / Logic
  Sources:
  • problems.ru web site

• Three Friends

  There are three friends, Weiss, Schwartzman, and Rotenberg. This trio of friends is special because one of them is blonde, one is ginger (redhead), and one has black hair. One day, the blonde guy said to Schwartzman: "We are a very special trio of friends! Notice that it's not only that the last names of the three of us mean colors, but also that each of our last names does not match his hair color."

  What hair color does each of these three people have?

  Note: "Schwartzman" means "black man", "Rotenberg" means "red mountain" and "Weiss" means "white" (from Yiddish).

  Topics:

  Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Number Tables Logic -> Truth-tellers and Liars Problems
  Sources:
  • problems.ru web site

• Corruption in Parliament

  The parliament of the magical country consists of `20` people. It is known that among the `20` members of parliament, there is at least one who is not corrupt. Additionally, it is known that for any two members of parliament we choose, one is necessarily corrupt. How many members of parliament of the magical country are corrupt?

  Topics:

  Algebra -> Inequalities Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Mitzi

  There are many cats named Mitzi. Which is more numerous: cats excluding the cats that are not named Mitzi, or animals named Mitzi excluding those that are not cats?

  Topics:

  Logic -> Reasoning / Logic Set Theory
  Sources:
  • problems.ru web site

• Clock Tower

  A bell in a clock tower rings `3` times in `12` seconds. How long does it take to ring `6` times?

  Topics:

  Arithmetic Algebra -> Word Problems Logic -> Reasoning / Logic Number Theory -> Division
  Sources:
  • problems.ru web site

• Question

  Yossi's mother told him: "All champions excel in their studies." Yossi replied: "I excel in my studies, therefore I am a champion!" Is Yossi's reasoning correct?

  Topics:

  Logic -> Reasoning / Logic
  Sources:
  • problems.ru web site

• The Restless Fly

  Cities A and B are 300 kilometers apart. Two cyclists start simultaneously from A and B, heading towards each other. Their speeds are constant and equal to `30` km/h and `20` km/h, respectively. At the same moment, a fly departs from city A, flying at a speed of `100` km/h. The fly overtakes the first cyclist and flies until it meets the cyclist who left from city B. The moment the fly meets the cyclist, it immediately turns around and flies back until it meets the first cyclist again, and then it turns around again, and so on, until the cyclists meet. How many kilometers did the fly travel?

  Topics:

  Arithmetic Algebra -> Equations Logic -> Reasoning / Logic Algebra -> Word Problems -> Motion Problems

• Question

  At `12:00`, the two hands of a clock coincide. In exactly how much time will they coincide again? How many times within a 24-hour period will they coincide?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Word Problems -> Motion Problems

• Question

  How long will it take a train traveling at a speed of `60` km/h and with a length of `50` meters to pass through a tunnel with a length of `50` meters?

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  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Word Problems -> Motion Problems Units of Measurement

• Playing with Chocolate

  Yossi and Danny are playing the following game. They have a chocolate bar of size `5xx7` squares, which is placed on a table. Each player, in turn, breaks one of the pieces of chocolate on the table along a straight line and returns the resulting pieces to the table. That is, in the first turn, the original chocolate bar is broken, and in the following turns, one piece is chosen from the pieces created up to that point and broken. It is only possible to break along the lines of the squares, and each break is from edge to edge. The player who cannot make a move loses. The winner eats all the chocolate.

  Which of the two can guarantee a victory for themselves, if Yossi makes the first move?

  Topics:

  Combinatorics -> Invariants Combinatorics -> Game Theory Logic -> Reasoning / Logic